Theorem finns ≪ | index | src | ≫

theorem finns (a: nat): $ finite a $;

Step	Hyp	Ref	Expression
1		finss	a C_ {x \| x < a} -> finite {x \| x < a} -> finite a
2		ssab2	A. x (x e. a -> x < a) <-> a C_ {x \| x < a}
3		ellt	x e. a -> x < a
4	3	ax_gen	A. x (x e. a -> x < a)
5	2, 4	mpbi	a C_ {x \| x < a}
6	1, 5	ax_mp	finite {x \| x < a} -> finite a
7		ltfin	finite {x \| x < a}
8	6, 7	ax_mp	finite a

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)